Editing Triangular bipyramid

{{Short description|Two tetrahedra joined by one face}}
{{good article}}
{{Infobox polyhedron
 | image = Triangular bipyramid.png
 | type =
 [[Bipyramid]]<br>[[Deltahedron|Deltahedra]]<br>[[Johnson solid|Johnson]]<br>
 {{math|[[gyroelongated pentagonal pyramid|''J''<sub>11</sub>]] – '''''J''<sub>12</sub>''' – [[Pentagonal bipyramid|''J''<sub>13</sub>]]}}<br>[[Noble polyhedron]]
 | faces = 6 [[triangle]]s
 | edges = 9
 | vertices = 5
 | vertex_config = <math> 3 \times (3^2) + 6 \times (3^2) </math>
 | symmetry = <math> D_{3 \mathrm{h}} </math>
 | dual = [[triangular prism]]
 | angle = As a Johnson solid:{{bulletlist
  | triangle-to-triangle: 70.5°
  | triangle-to-triangle if tetrahedrons being attached: 141.1°
  }}
 | properties = [[Convex set|convex]],<br>[[composite polyhedron|composite]],<br>[[face-transitive]]
}}
A '''triangular bipyramid''' is a [[hexahedron]] with six triangular faces constructed by attaching two [[Tetrahedron|tetrahedra]] face-to-face. The same shape is also known as a '''triangular dipyramid'''{{r|trigg|rajwade}} or '''trigonal bipyramid'''.{{r|king}} If these tetrahedra are regular, all faces of a triangular bipyramid are [[Equilateral triangle|equilateral]]. It is an example of a [[deltahedron]], [[composite polyhedron]], and [[Johnson solid]].

Many polyhedra are related to the triangular bipyramid, such as similar shapes derived from different approaches and the [[triangular prism]] as its [[dual polyhedron]]. Applications of a triangular bipyramid include [[trigonal bipyramidal molecular geometry]] which describes its [[atom cluster]], a solution of the [[Thomson problem]], and the representation of [[Color model|color order systems]] by the eighteenth century.

== Special cases ==
=== As a right bipyramid ===
Like other [[bipyramid]]s, a triangular bipyramid can be constructed by attaching two tetrahedra face-to-face.{{r|rajwade}} These tetrahedra cover their triangular base, and the resulting polyhedron has six triangles, five [[Vertex (geometry)|vertices]], and nine edges.{{r|king}} A triangular bipyramid is said to be ''right'' if the tetrahedra are symmetrically regular and both of their [[Apex (geometry)|apices]] are on a line passing through the center of the base; otherwise, it is ''oblique''.{{r|niu-xu|alexandrov}}

[[File:Graph of triangular bipyramid.svg|Graph of a triangular bipyramid|alt=A line drawing with multicolored dots|thumb|left|upright]]
According to [[Steinitz's theorem]], a [[Graph (discrete mathematics)|graph]] can be represented as the [[n-skeleton|skeleton]] of a polyhedron if it is a [[Planar graph|planar]] (the edges of the graph do not cross, but intersect at the point) and [[k-vertex-connected graph|three-connected graph]] (one of any two vertices leaves a connected subgraph when removed). A triangular bipyramid is represented by a graph with nine edges, constructed by adding one vertex to the vertices of a [[wheel graph]] representing [[tetrahedra]].{{r|tutte|ssp}}

Like other right bipyramids, a triangular bipyramid has [[Point groups in three dimensions|three-dimensional point-group symmetry]], the [[dihedral group]] <math> D_{3 \mathrm{h}} </math> of order twelve: the appearance of a triangular bipyramid is unchanged as it rotated by one-, two-thirds, and full angle around the [[Axial symmetry|axis of symmetry]] (a line passing through two vertices and the base's center vertically), and it has [[mirror symmetry]] with any bisector of the base; it is also symmetrical by reflection across a horizontal plane.{{r|ak}} A triangular bipyramid is [[face-transitive]] (or isohedral).{{r|mclean}}
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=== As a Johnson solid ===
{{multiple image
 | image1 = Triangular dipyramid.png
 | alt1 = A triangular bipyramid with regular faces
 | image2 = Triangular bipyramid (symmetric net).svg
 | alt2 = Multicolor, flat image of a triangular bipyramid
 | footer = Triangular bipyramid with regular faces alongside its [[Net (polyhedron)|net]]
 | total_width = 400
}}
[[File:J12 triangular bipyramid.stl|thumb|alt=A grayscale image|3D model of a triangular bipyramid as a Johnson solid]]
If the tetrahedra are regular, all edges of a triangular bipyramid are equal in length and form [[Equilateral triangle|equilateral triangular]] faces. A polyhedron with only equilateral triangles as faces is called a [[deltahedron]]. There are eight convex deltahedra, one of which is a triangular bipyramid with [[regular polygon]]al faces.{{r|trigg}} A convex polyhedron in which all of its faces are regular polygons is the [[Johnson solid]], and every convex deltahedron is a Johnson solid. A triangular bipyramid with regular faces is numbered as the twelfth Johnson solid <math> J_{12} </math>.{{r|uehara}} It is an example of a [[composite polyhedron]] because it is constructed by attaching two [[Tetrahedron|regular tetrahedra]].{{r|timofeenko-2009|berman}}

A triangular bipyramid's surface area <math> A </math> is six times that of each triangle. Its volume <math> V </math> can be calculated by slicing it into two tetrahedra and adding their volume. In the case of edge length <math> a </math>, this is:{{r|berman}}
<math display="block"> \begin{align}
 A &= \frac{3\sqrt{3}}{2}a^2 &\approx 2.598a^2, \\
 V &= \frac{\sqrt{2}}{6}a^3 &\approx 0.238a^3.
\end{align} </math>

The [[dihedral angle]] of a triangular bipyramid can be obtained by adding the dihedral angle of two regular tetrahedra. The dihedral angle of a triangular bipyramid between adjacent triangular faces is that of the regular tetrahedron: 70.5 degrees. In an edge where two tetrahedra are attached, the dihedral angle of adjacent triangles is twice that: 141.1 degrees.{{r|johnson}}
{{-}}

== Related polyhedra ==
[[File:GoldnerHararyJmol2C.jpg|alt=Geometric realization of the Goldner–Harary graph|The Goldner–Harary graph represents a triangular bipyramid, augmented by tetrahedra.|thumb|upright]]
Some types of triangular bipyramids may be derived in different ways. The [[Kleetope]] of a polyhedron is a construction involving the attachment of pyramids. A triangular bipyramid's Kleetope can be constructed from a triangular bipyramid by attaching tetrahedra to each of its faces, replacing them with three other triangles; the skeleton of the resulting polyhedron represents the [[Goldner–Harary graph]].{{r|grunbaum|ewald}} Another type of triangular bipyramid results from cutting off its vertices, a process known as [[Truncation (geometry)|truncation]].{{r|hceg}}

Bipyramids are the [[dual polyhedron]] of [[Prism (geometry)|prisms]]. This means the bipyramids' vertices correspond to the faces of a prism, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other; doubling it results in the original polyhedron. A triangular bipyramid is the dual polyhedron of a [[triangular prism]], and vice versa.{{r|sibley|king}} A triangular prism has five faces, nine edges, and six vertices, with the same symmetry as a triangular bipyramid.{{r|king}}

== Applications ==
[[File:N 2 to 5 ThomsonSolutions.png|The known solution of the Thomson problem, with one a triangular bipyramid|alt=Four circles, with geometric figures inside them|thumb|upright]]
The [[Thomson problem]] concerns the minimum energy configuration of charged particles on a sphere. A triangular bipyramid is a known solution in the case of five electrons, placing vertices of a triangular bipyramid [[Circumscribed sphere|within a sphere]].{{r|shdc}} This solution is aided by a mathematically rigorous computer.{{r|schwartz}}

A [[chemical compound]]'s [[trigonal bipyramidal molecular geometry]] may be described as the [[atom cluster]] of a triangular bipyramid. This molecule has a [[main-group element]] without an active [[lone pair]], described by a model which predicts the geometry of molecules known as [[VSEPR theory]].{{r|phf}} Examples of this structure include [[phosphorus pentafluoride]] and [[phosphorus pentachloride]] in the gaseous [[Phase (matter)|phase]].{{r|housecroft}}

In [[color theory]], the triangular bipyramid was used to represent the three-dimensional [[Primary color#Color order systems|color-order system in primary colors]]. German astronomer [[Tobias Mayer]] wrote in 1758 that each of its vertices represents a color: white and black are the top and bottom axial vertices, respectively, and the rest of the vertices are red, blue, and yellow.{{r|kuehni-2003|kuehni-2013}}
{{-}}

== References ==
{{reflist|refs=

<ref name="ak">{{cite book
 | last1 = Alexander | first1 = Daniel C.
 | last2 = Koeberlin | first2 = Geralyn M.
 | year = 2014
 | title = Elementary Geometry for College Students
 | url = https://books.google.com/books?id=EN_KAgAAQBAJ&pg=PA403
 | edition = 6th
 | publisher = Cengage Learning
 | page = 403
 | isbn = 978-1-285-19569-8
}}</ref>

<ref name="alexandrov">{{cite journal
 | last = Alexandrov | first = Victor
 | year = 2017
 | title = How many times can the volume of a convex polyhedron be increased by isometric deformations?
 | journal = Beiträge zur Algebra und Geometrie
 | volume = 58
 | issue = 3
 | pages = 549–554
 | doi = 10.1007/s13366-017-0336-8
 | arxiv = 1607.06604
}}</ref>

<ref name="berman">{{cite journal
 | last = Berman | first = Martin
 | doi = 10.1016/0016-0032(71)90071-8
 | journal = Journal of the Franklin Institute
 | mr = 290245
 | pages = 329–352
 | title = Regular-faced convex polyhedra
 | volume = 291
 | year = 1971| issue = 5
 }}</ref>

<ref name="ewald">{{cite journal
 | last = Ewald | first = Günter
 | year = 1973
 | title = Hamiltonian circuits in simplicial complexes
 | journal = Geometriae Dedicata
 | volume = 2
 | issue = 1
 | pages = 115–125
 | doi = 10.1007/BF00149287
 | s2cid = 122755203
}}</ref>

<ref name="grunbaum">{{cite book
 | last = Grünbaum | first = Branko | authorlink = Branko Grünbaum
 | year = 1967
 | title = Convex Polytopes | title-link = Convex Polytopes
 | publisher = Wiley Interscience
 | page = 357
}}. Same page, 2nd ed., Graduate Texts in Mathematics 221, Springer-Verlag, 2003, {{ISBN|978-0-387-40409-7}}.</ref>

<ref name="hceg">{{cite journal
 | last1 = Haji-Akbari | first1 = Amir
 | last2 = Chen | first2 = Elizabeth R.
 | last3 = Engel | first3 = Michael
 | last4 = Glotzer | first4 = Sharon C.
 | arxiv = 1304.3147
 | journal = Phys. Rev. E
 | page = 012127
 | title = Packing and self-assembly of truncated triangular bipyramids
 | volume = 88
 | year = 2013 | issue = 1
 | doi=10.1103/physreve.88.012127| pmid = 23944434
 | bibcode = 2013PhRvE..88a2127H| s2cid = 8184675
 }}.</ref>

<ref name="housecroft">{{cite book
 | last1 = Housecroft | first1 = C. E.
 | last2 = Sharpe | first2 = A. G.
 | year = 2004
 | title = Inorganic Chemistry
 | publisher = Prentice Hall
 | edition = 2nd
 | isbn = 978-0-13-039913-7
 | page = 407
}}</ref>

<ref name="johnson">{{cite journal
 | last = Johnson | first = Norman W. | author-link = Norman W. Johnson
 | year = 1966
 | title = Convex polyhedra with regular faces
 | journal = [[Canadian Journal of Mathematics]]
 | volume = 18
 | pages = 169–200
 | doi = 10.4153/cjm-1966-021-8
 | mr = 0185507
 | s2cid = 122006114
 | zbl = 0132.14603
 | doi-access = free
}}</ref>

<ref name="king">{{cite book
 | last = King | first = Robert B.
 | editor-last1 = Bonchev | editor-first1 = Danail D.
 | editor-last2 = Mekenyan | editor-first2 = O.G.
 | year = 1994
 | contribution = Polyhedral Dynamics
 | contribution-url = https://books.google.com/books?id=c3fsCAAAQBAJ&pg=PA113
 | title = Graph Theoretical Approaches to Chemical Reactivity
 | publisher = Springer
 | doi = 10.1007/978-94-011-1202-4
 | isbn = 978-94-011-1202-4
}}</ref>

<ref name="kuehni-2003">{{cite book
 | last = Kuehni | first = Rolf G.
 | year = 2003
 | title = Color Space and Its Divisions: Color Order from Antiquity to the Present
 | page = 53
 | publisher = John & Sons Wiley
 | isbn = 978-0-471-46146-3
 | url = https://books.google.com/books?id=2kFVSRGC650C&pg=PA53
}}</ref>

<ref name="kuehni-2013">{{cite book
 | last = Kuehni | first = Rolf G.
 | year = 2013
 | title = Color: An Introduction to Practice and Principles
 | page = 198
 | publisher = John & Sons Wiley
 | url = https://books.google.com/books?id=OD0ogyBwvgwC&pg=PA198
 | isbn = 978-1-118-17384-8
}}</ref>

<ref name="mclean">{{cite journal
 | last = McLean | first = K. Robin
 | year = 1990
 | title = Dungeons, dragons, and dice
 | journal = The Mathematical Gazette
 | volume = 74 | issue = 469 | pages = 243–256
 | doi = 10.2307/3619822
 | jstor = 3619822
 | s2cid = 195047512
}}</ref>

<ref name="niu-xu">{{cite journal
 | last1 = Niu | first1 = Wenxin
 | last2 = Xu | first2 = Guobao
 | year = 2011
 | title = Crystallographic control of noble metal nanocrystals
 | journal = Nano Today
 | volume = 6
 | issue = 3
 | pages = 265–285
 | doi = 10.1016/j.nantod.2011.04.006
}}</ref>

<ref name="phf">{{cite book
 | last1 = Petrucci | first1 = R. H.
 | last2 = Harwood | first2 = W. S.
 | last3 = Herring | first3 = F. G.
 | year = 2002
 | title = General Chemistry: Principles and Modern Applications
 | publisher = Prentice-Hall
 | edition = 8th
 | pages = 413–414
 | isbn = 978-0-13-014329-7
}} See table 11.1.</ref>

<ref name="rajwade">{{cite book
 | last = Rajwade | first = A. R.
 | title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem
 | series = Texts and Readings in Mathematics
 | year = 2001
 | url = https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84
 | page = 84
 | publisher = Hindustan Book Agency
 | isbn = 978-93-86279-06-4
 | doi = 10.1007/978-93-86279-06-4
}}</ref>

<!-- <ref name="rw">{{cite book
 | last1 = Rooney | first1 = Joe
 | last2 = Wilson | first2 = Robin J.
 | year = 1984
 | contribution = The mobility of a graph
 | editor-last1 = Koh | editor-first1 = Khee Meng
 | editor-last2 = Yap | editor-first2 = Hian Poh
 | title = Graph Theory Singapore 1983: Proceedings of the First Southeast Asian Graph Theory Colloquium, Held in Singapore, May 10-28, 1983
 | url = https://books.google.com/books?id=82t8CwAAQBAJ&pg=PA148
 | page = 148
 | publisher = Springer
 | doi = 10.1007/BFb0073099
 | isbn = 978-3-540-38924-8
}}</ref>-->

<ref name="schwartz">{{cite journal
 | last = Schwartz | first = Richard Evan
 | year = 2013
 | title = The Five-Electron Case of Thomson's Problem
 | journal = Experimental Mathematics
 | volume = 22
 | issue = 2
 | pages = 157–186
 | doi = 10.1080/10586458.2013.766570
| s2cid = 38679186
 }}</ref>

<ref name="shdc">{{citation
 | last1 = Sloane | first1 = N. J. A. | author1-link = Neil Sloane
 | last2 = Hardin | first2 = R. H.
 | last3 = Duff | first3 = T. D. S.
 | last4 = Conway | first4 = J. H. | author4-link = John Horton Conway
 | year = 1995
 | title = Minimal-energy clusters of hard spheres
 | journal = [[Discrete & Computational Geometry]]
 | volume = 14
 | issue = 3
 | pages = 237–259
 | doi = 10.1007/BF02570704
 | mr = 1344734
 | s2cid = 26955765 | doi-access = free
}}</ref>

<ref name="sibley">{{cite book
 | last = Sibley | first = Thomas Q.
 | year = 2015
 | title = Thinking Geometrically: A Survey of Geometries
 | publisher = Mathematical Association of American
 | page = 53
 | url = https://books.google.com/books?id=EUh2CgAAQBAJ&pg=PA53
 | isbn =978-1-939512-08-6
}}</ref>

<ref name="ssp">{{cite journal
 | last1 = Sajjad | first1 = Wassid
 | last2 = Sardar | first2 = Muhammad S.
 | last3 = Pan | first3 = Xiang-Feng
 | year = 2024
 | title = Computation of resistance distance and Kirchhoff index of chain of triangular bipyramid hexahedron
 | journal = Applied Mathematics and Computation
 | volume = 461
 | pages = 1–12
 | doi = 10.1016/j.amc.2023.128313
 | s2cid = 261797042
}}</ref>

<ref name="timofeenko-2009">{{cite journal
 | last = Timofeenko | first = A. V.
 | year = 2009
 | title = Convex Polyhedra with Parquet Faces
 | journal = Docklady Mathematics
 | url = https://www.interocitors.com/tmp/papers/timo-parquet.pdf
 | volume = 80 | issue = 2
 | pages = 720–723
 | doi = 10.1134/S1064562409050238
}}</ref>

<ref name="trigg">{{cite journal
 | last = Trigg | first = Charles W.
 | year = 1978
 | title = An infinite class of deltahedra
 | journal = Mathematics Magazine
 | volume = 51
 | issue = 1
 | pages = 55–57
 | doi = 10.1080/0025570X.1978.11976675
 | jstor = 2689647
 | mr = 1572246
}}</ref>

<ref name="tutte">{{cite book
 | last = Tutte | first = W. T.
 | year = 2001
 | title = Graph Theory
 | url = https://books.google.com/books?id=uTGhooU37h4C&pg=PA113
 | publisher = Cambridge University Press
 | page = 113
| isbn = 978-0-521-79489-3
 }}</ref>

<ref name="uehara">{{cite book
 | last = Uehara | first = Ryuhei
 | year = 2020
 | title = Introduction to Computational Origami: The World of New Computational Geometry
 | url = https://books.google.com/books?id=51juDwAAQBAJ
 | publisher = Springer
 | isbn = 978-981-15-4470-5
 | doi = 10.1007/978-981-15-4470-5
| s2cid = 220150682
 }}</ref>

,<!--<ref name="wohlleben">{{cite conference
 | last = Wohlleben | first = Eva
 | editor-last = Cocchiarella | editor-first = Luigi
 | year = 2019
 | contribution = Duality in Non-Polyhedral Bodies Part I: Polyliner
 | conference = International Conference on Geometry and Graphics
 | title = ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics: 40th Anniversary - Milan, Italy, August 3-7, 2018
 | url = https://books.google.com/books?id=rEpjDwAAQBAJ
 | publisher = Springer
 | isbn = 978-3-319-95588-9
 | doi = 10.1007/978-3-319-95588-9
}}</ref>-->

}}

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